Accurate image segmentation is crucial for medical imaging applications. The prevailing deep learning approaches typically rely on very large training datasets with high-quality manual annotations, which are often not available in medical imaging. We introduce Annotation-effIcient Deep lEarning (AIDE) to handle imperfect datasets with an elaborately designed cross-model self-correcting mechanism. AIDE improves the segmentation Dice scores of conventional deep learning models on open datasets possessing scarce or noisy annotations by up to 30%. For three clinical datasets containing 11,852 breast images of 872 patients from three medical centers, AIDE consistently produces segmentation maps comparable to those generated by the fully supervised counterparts as well as the manual annotations of independent radiologists by utilizing only 10% training annotations. Such a 10-fold improvement of efficiency in utilizing experts labels has the potential to promote a wide range of biomedical applications.